A man 8 friends whom he wants to invite for dinner. The number of ways

Question

A man has 8 friends whom he wants to invite to dinner. In how many ways can he invite at least one of them?

A. 8
B. 255
C. 8! - 1
D. 256
E. 7

Source :Quantpdf

A man 8 friends whom he wants to invite for dinner. The number of ways in which he can invite at least one of them is

BrentGMATPrepNow · Accepted Answer

Take the task of inviting friends and break it into   stages  .

ASIDE: Let's let A, B, C, D, E, F, G and H represent the 8 friends

Stage 1 : Decide whether or not to invite friend A 
You have 2 options: invite friend A or don't invite friend A
So, we can complete stage 1 in  2  ways

Stage 2 : Decide whether or not to invite friend B 
You have 2 options: invite friend B or don't invite friend B
So, we can complete stage 2 in  2  ways

Stage 3 : Decide whether or not to invite friend C 
So, we can complete stage 3 in  2  ways 
.
.
.
 Stage 8 : Decide whether or not to invite friend H 
So, we can complete stage 8 in  2  ways

By the Fundamental Counting Principle (FCP), we can complete all 8 stages (and create a guest list) in  (2)(2)(2)(2)(2)(2)(2)(2)  ways (= 256 ways)

NOTE: in these calculations, one of the possible outcomes is that ZERO friends are invited. The question says that AT LEAST ONE friend must come. 
So, we must subtract this 1 outcome from our solution.

So, total number of ways to invite friends = 256 - 1 = 255

Answer:  B

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So, be sure to learn it.

RELATED VIDEOS
 https://www.youtube.com/watch?v=w5d5bvCRyd4

https://www.youtube.com/watch?v=_PFBp2BoMUY

https://www.youtube.com/watch?v=05a0A7vjG8I

BrentGMATPrepNow · Answer

Another approach (for Mo2men :-D

) Number of ways to invite at least 1 friend = (# of ways to invite exactly 1 friend) + (# of ways to invite exactly 2 friends) + (# of ways to invite exactly 3 friends) + (# of ways to invite exactly 4 friends) + . . . . + (# of ways to invite exactly 8 friends) Since the order in which we invite the friends does not matter, we can use COMBINATIONS. For example, the number of ways to invite exactly 2 friends = 8C2 And the number of ways to invite exactly 3 friends = 8C3 etc. So, the number of ways to invite at least 1 friend = 8C1 + 8C2 + 8C3 + . . . + 8C7 + 8C8 = 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 = 255 RELATED VIDEO https://www.youtube.com/watch?v=05a0A7vjG8I

Mo2men · Answer

Hi Brent,

Can you show another approach?

What 'at least' mean?

BrentGMATPrepNow · Answer

"at least" means "greater than or equal to"

So, if I say that I own at least 3 guitars, then the number of guitars I own = 3 or 4 or 5 or 6....

Cheers,
Brent

ScottTargetTestPrep · Answer

We can use the following equation:

Total number of ways to invite friends = number of ways to invite at least one friend - number of ways to invite zero friends.

The total number of ways the man can invite his friends is 2^8 since he can or cannot invite each friend. The number of ways to invite no friends is 1. Thus, the number of ways to invite at least one friend is 2^8 - 1 = 256 - 1 = 255.

Answer: B

kablayi · Answer

The question could be interpreted in another way. "If we will flip a coin 8 times what is the number of ways we get at least one tail"

When you flip a coin 8 times there are 2^8 different ways the result may come out. One of the possible results is to get 8 heads - HHHHHHHH. Now you can imagine tail is to invite a friend and head is not to invite. Getting 8 heads means not to invite any of them.

Therefore, all outcomes satisfy us except when we get eight heads. There is only one way to get 8 heads. So we deduct this one outcome from total number of possible outcomes

Pritishd · Answer

An alternate approach for this question would be to find the subsets of a set with 8 entities.

Lets take a smaller set, say there were only 3 friends (A, B & C) then the ways to invite at least 1 would be A, B, C, AB, AC, BC and ABC (7 ways). This is  2^n-1      in terms of the formula to find the subset of a set with n entities. We subtracted 1 because  2^n  also includes an empty set, whereas we have been provided a constraint of at least 1.

Applying the same to the problem at hand, there are 8 friends, hence  2^8-1 = 256-1 = 255  (Ans B)

Kinshook · Answer

Given:  A man 8 friends whom he wants to invite for dinner.

Asked:  The number of ways in which he can invite at least one of them is

No of ways to invite at least 1 friends =  ^8C_1 + ^8C_2 + ^8C_3 + ^8C_4 + ^8C_5 + ^8C_6 + ^8C_7 + ^8C_8

(1+x)^n = ^nC_0 x^0 + ^nC_1 x + ^nC_2 x^2 + ... + ^nC_r x^r +.... + ^nC_n x^n

n =8 
 (1+x)^8 = ^8C_0 x^0 + ^8C_1 x + ^8C_2 x^2 + ... + ^8C_r x^r +.... + ^8C_8 x^n 
When x =1
 2^8 = 1 + ^8C_1 + ^8C_2 + ^8C_3 + ^8C_4 + ^8C_5 + ^8C_6 + ^8C_7 + ^8C_8 
 2^8 -1= ^8C_1 + ^8C_2 + ^8C_3 + ^8C_4 + ^8C_5 + ^8C_6 + ^8C_7 + ^8C_8 = 256-1 =255

IMO B

Amazingakhil · Answer

Each of the 8 friends can either be invited or not invited → 2 choices each.
Total possible invitation combinations = 2^8
=256.
This includes the case where no one is invited.
Since at least one must be invited, subtract 1 → 256−1=255.
Answer: (B) 255

Adit_ · Answer

Bunuel  Kindly help correcting the grammar of the stem.

Bunuel · Answer

__________________
Done. Thank you!

A man 8 friends whom he wants to invite for dinner. The number of ways

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

A man 8 friends whom he wants to invite for dinner. The number of ways

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards